逆多项式的Chebyshev级数展开
Chebyshev Series Expansion of Inverse Polynomials
摘要：
逆多项式具有切比雪夫级数展开式1/ \sum（j=0. k）b_j*T_j（x）= \sum“（n=0.. oo）a_n*T_n（x），如果多项式在[-1，1]中没有根。如果逆多项式被分解成部分分式，则a_n是多项式根的简单函数的线性组合。如果系数a_n的第一个k是已知的，则其它系数变为这些系数的线性组合，其中展开系数从b_j递归地导出。在一个密切相关的主题上，找到一个对给定f（x）具有最小相对误差的多项式近似等于找到f（x）/sum_（j=0）中的b_j。k）b_j*T_j（x）=1+sum_（n=k+1. oo）a_n*T_n（x），并且可以用牛顿方法处理，只要f（x）的切比雪夫展开是已知的。
英文摘要：
An inverse polynomial has a Chebyshev series expansion 1/\sum(j=0..k)b_j*T_j(x)=\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no roots in [-1,1]. If the inverse polynomial is decomposed into partial fractions, the a_n are linear combinations of simple functions of the polynomial roots. If the first k of the coefficients a_n are known, the others become linear combinations of these with expansion coefficients derived recursively from the b_j's. On a closely related theme, finding a polynomial with minimum relative error towards a given f(x) is approximately equivalent to finding the b_j in f(x)/sum_(j=0..k)b_j*T_j(x)=1+sum_(n=k+1..oo) a_n*T_n(x), and may be handled with a Newton method providing the Chebyshev expansion of f(x) is known.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
